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Theorem cfss 10201
Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfss.1 𝐴 ∈ V
Assertion
Ref Expression
cfss (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cfss.1 . . . . . 6 𝐴 ∈ V
21cflim3 10198 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
3 fvex 6855 . . . . . . 7 (card‘𝑥) ∈ V
43dfiin2 4994 . . . . . 6 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5 cardon 9880 . . . . . . . . . 10 (card‘𝑥) ∈ On
6 eleq1 2825 . . . . . . . . . 10 (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
75, 6mpbiri 257 . . . . . . . . 9 (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
87rexlimivw 3148 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
98abssi 4027 . . . . . . 7 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10 limuni 6378 . . . . . . . . . . . 12 (Lim 𝐴𝐴 = 𝐴)
1110eqcomd 2742 . . . . . . . . . . 11 (Lim 𝐴 𝐴 = 𝐴)
12 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1312eqcomd 2742 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
1413biantrud 532 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15 unieq 4876 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 𝑥 = 𝐴)
1615eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 𝐴 = 𝐴))
171pwid 4582 . . . . . . . . . . . . . . . . 17 𝐴 ∈ 𝒫 𝐴
18 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴𝐴 ∈ 𝒫 𝐴))
1917, 18mpbiri 257 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴)
2019biantrurd 533 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2116, 20bitr3d 280 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2221anbi1d 630 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
2314, 22bitr2d 279 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ 𝐴 = 𝐴))
241, 23spcev 3565 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2511, 24syl 17 . . . . . . . . . 10 (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26 df-rex 3074 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27 rabid 3427 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
2827anbi1i 624 . . . . . . . . . . . 12 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2928exbii 1850 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3026, 29bitri 274 . . . . . . . . . 10 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3125, 30sylibr 233 . . . . . . . . 9 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32 fvex 6855 . . . . . . . . . 10 (card‘𝐴) ∈ V
33 eqeq1 2740 . . . . . . . . . . 11 (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
3433rexbidv 3175 . . . . . . . . . 10 (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
3532, 34spcev 3565 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3631, 35syl 17 . . . . . . . 8 (Lim 𝐴 → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37 abn0 4340 . . . . . . . 8 ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3836, 37sylibr 233 . . . . . . 7 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39 onint 7725 . . . . . . 7 (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
409, 38, 39sylancr 587 . . . . . 6 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
414, 40eqeltrid 2842 . . . . 5 (Lim 𝐴 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
422, 41eqeltrd 2838 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43 fvex 6855 . . . . 5 (cf‘𝐴) ∈ V
44 eqeq1 2740 . . . . . 6 (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
4544rexbidv 3175 . . . . 5 (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
4643, 45elab 3630 . . . 4 ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
4742, 46sylib 217 . . 3 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48 df-rex 3074 . . 3 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
4947, 48sylib 217 . 2 (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50 simprl 769 . . . . . . . 8 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴})
5150, 27sylib 217 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
5251simpld 495 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
5352elpwid 4569 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥𝐴)
54 simpl 483 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
56 limord 6377 . . . . . . . . . . . 12 (Lim 𝐴 → Ord 𝐴)
57 ordsson 7717 . . . . . . . . . . . 12 (Ord 𝐴𝐴 ⊆ On)
5856, 57syl 17 . . . . . . . . . . 11 (Lim 𝐴𝐴 ⊆ On)
59 sstr 3952 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
6058, 59sylan2 593 . . . . . . . . . 10 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61 onssnum 9976 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
6255, 60, 61sylancr 587 . . . . . . . . 9 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63 cardid2 9889 . . . . . . . . 9 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
6462, 63syl 17 . . . . . . . 8 ((𝑥𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
6564ensymd 8945 . . . . . . 7 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
6653, 54, 65syl2anc 584 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67 simprr 771 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
6866, 67breqtrrd 5133 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
6951simprd 496 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 = 𝐴)
7053, 68, 693jca 1128 . . . 4 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
7170ex 413 . . 3 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7271eximdv 1920 . 2 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7349, 72mpd 15 1 (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   cint 4907   ciin 4955   class class class wbr 5105  dom cdm 5633  Ord word 6316  Oncon0 6317  Lim wlim 6318  cfv 6496  cen 8880  cardccrd 9871  cfccf 9873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-er 8648  df-en 8884  df-dom 8885  df-card 9875  df-cf 9877
This theorem is referenced by: (None)
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